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\begin{document}

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\section*{leaf of the distribution}

An Introduction to Distributions and Foliations

MTH 864 · Spring 2008 In smooth manifold theory, the notion of a tangent space makes it possi- ble for differentiation to take place on an abstract manifold. In this paper, the notion of a distribution will be presented which makes it possible for in- tegration to take place on an abstract manifold. The first section introduces terminology and builds intuition via an analogy to the concept of integral curves. The second section presents the Frobenius theorem–one of the foun- dational results in smooth manifold theory. The concluding section leaves the reader with foliations and a brief look at their connection to the Frobenius theorem. 1 Preliminaries Let M be an m-dimensional manifold, TpM the tangent space to p ∈ M, and TM the tangent bundle of M. Throughout this work, things are implicitly smooth. Recall that a vector field on M is a section of TM and can be thought of as a choice of tangent vector at every point in the manifold. Given a vector field V , one can consider an associated integral curve. Definition. An integral curve of a vector field V is a curve γ : (a, b) → M (a and b real numbers) such that γ ′ (t) = Vγ(t) for all t ∈ (a, b).

1Definition. The Lie bracket of the vector fields V and W is V, W = V W − W V . It is well known that V, W itself defines a (natural) vector field. What we will find below is that “nice” means given any vector fields V , W with Vp, Wp ∈ Dp for all p in some neighborhood U, it follows that Vp, Wp ∈ Dp. If this condition is satisfied (it is only necessary to check it on the basis vector fields), then the distribution D is called involutive. Lemma. If D is an integrable distribution, then D is necessarily involutive. Proof. Let V and W be local sections of D ⊂ TM defined on some U ⊂ M and let p ∈ U. Since D is integrable, we have some S an integral manifold of D containing p. Since V and W are sections of D, we know that V and W are tangent to S on U. By properties of the naturality of the Lie bracket, V, W is also tangent to S and so V, Wp ∈ Dp. This holds for all points in U and so D is involutive. As an example, consider the 2-dimensional distribution D on R

Turka Tachloo

3 with basis vector fields

\begin{verbatim}                                   writer:
                                   NADEEM AHMAD RATHER 
                                   M.PHIL MATHEMATICS
                                 R/O TURKA TACHLOO
                              ANANTNAG(JAMMU AND KASHMIR)\end{verbatim}


\end{document}